Interpolation & Polynomial Approximation. Cubic Spline Interpolation II
|
|
|
- Adam Walton
- 5 years ago
- Views:
Transcription
1 Interpolation & Polynomial Approximation Cubic Spline Interpolation II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011 Brooks/Cole, Cengage Learning
2 Outline 1 Unique natural cubic spline interpolant Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 2 / 29
3 Outline 1 Unique natural cubic spline interpolant 2 Natural cubic spline approximating f (x) = e x Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 2 / 29
4 Outline 1 Unique natural cubic spline interpolant 2 Natural cubic spline approximating f (x) = e x 3 Natural cubic spline approximating 3 0 ex dx Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 2 / 29
5 Outline 1 Unique natural cubic spline interpolant 2 Natural cubic spline approximating f (x) = e x 3 Natural cubic spline approximating 3 0 ex dx Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 3 / 29
6 Existence of a unique natural spline interpolant Theorem Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 4 / 29
7 Existence of a unique natural spline interpolant Theorem If f is defined at a = x 0 < x 1 < < x n = b, then f has a unique natural spline interpolant S on the nodes x 0, x 1,..., x n ; that is, a spline interpolant that satisfies the natural boundary conditions S (a) = 0 and S (b) = 0 Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 4 / 29
8 Existence of a unique natural spline interpolant Proof (1/4) Using the notation S j (x) = a j + b j (x x j ) + c j (x x j ) 2 + d j (x x j ) 3 the boundary conditions in this case imply that c n = 1 2 S n(x n )2 = 0 and that 0 = S (x 0 ) = 2c 0 + 6d 0 (x 0 x 0 ) so c 0 = 0. Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 5 / 29
9 Existence of a unique natural spline interpolant Proof (1/4) Using the notation S j (x) = a j + b j (x x j ) + c j (x x j ) 2 + d j (x x j ) 3 the boundary conditions in this case imply that c n = 1 2 S n(x n )2 = 0 and that 0 = S (x 0 ) = 2c 0 + 6d 0 (x 0 x 0 ) so c 0 = 0. The two equations c 0 = 0 and c n = 0 together with the equations h j 1 c j 1 + 2(h j 1 + h j )c j + h j c j+1 = 3 h j (a j+1 a j ) 3 h j 1 (a j a j 1 ) produce a linear system described by the vector equation Ax = b: Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 5 / 29
10 Existence of a unique natural spline interpolant Proof (2/4) A is the (n + 1) (n + 1) matrix: h 0 2(h 0 + h 1 ) h A = 0 h 1 2(h 1 + h 2 ) h hn 2 2(h n 2 + h n 1 ) h n Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 6 / 29
11 Existence of a unique natural spline interpolant Proof (3/4) b and x are the vectors 0 3 h 1 (a 2 a 1 ) 3 h 0 (a 1 a 0 ) b =. 3 h n 1 (a n a n 1 ) 3 h n 2 (a n 1 a n 2 ) 0 c 0 c 1 and x =. c n Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 7 / 29
12 Existence of a unique natural spline interpolant Proof (4/4) The matrix A is strictly diagonally dominant, that is, in each row the magnitude of the diagonal entry exceeds the sum of the magnitudes of all the other entries in the row.
13 Existence of a unique natural spline interpolant Proof (4/4) The matrix A is strictly diagonally dominant, that is, in each row the magnitude of the diagonal entry exceeds the sum of the magnitudes of all the other entries in the row. A linear system with a matrix of this form can be shown Theorem to have a unique solution for c 0, c 1,..., c n.
14 Existence of a unique natural spline interpolant Proof (4/4) The matrix A is strictly diagonally dominant, that is, in each row the magnitude of the diagonal entry exceeds the sum of the magnitudes of all the other entries in the row. A linear system with a matrix of this form can be shown Theorem to have a unique solution for c 0, c 1,..., c n. The solution to the cubic spline problem with the boundary conditions S (x 0 ) = S (x n ) = 0 can be obtained by applying the Natural Cubic Spline Algorithm.
15 Natural Cubic Spline Algorithm To construct the cubic spline interpolant S for the function f, defined at the numbers x 0 < x 1 < < x n, satisfying S (x 0 ) = S (x n ) = 0 (Note: S(x) = S j (x) = a j + b j (x x j ) + c j (x x j ) 2 + d j (x x j ) 3 for x j x x j+1.): Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 9 / 29
16 Natural Cubic Spline Algorithm To construct the cubic spline interpolant S for the function f, defined at the numbers x 0 < x 1 < < x n, satisfying S (x 0 ) = S (x n ) = 0 (Note: S(x) = S j (x) = a j + b j (x x j ) + c j (x x j ) 2 + d j (x x j ) 3 for x j x x j+1.): INPUT n; x 0, x 1,..., x n ; a 0 = f (x 0 ), a 1 = f (x 1 ),..., a n = f (x n ) Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 9 / 29
17 Natural Cubic Spline Algorithm To construct the cubic spline interpolant S for the function f, defined at the numbers x 0 < x 1 < < x n, satisfying S (x 0 ) = S (x n ) = 0 (Note: S(x) = S j (x) = a j + b j (x x j ) + c j (x x j ) 2 + d j (x x j ) 3 for x j x x j+1.): INPUT n; x 0, x 1,..., x n ; a 0 = f (x 0 ), a 1 = f (x 1 ),..., a n = f (x n ) OUTPUT a j, b j, c j, d j for j = 0, 1,..., n 1 Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 9 / 29
18 Natural Cubic Spline Algorithm To construct the cubic spline interpolant S for the function f, defined at the numbers x 0 < x 1 < < x n, satisfying S (x 0 ) = S (x n ) = 0 (Note: S(x) = S j (x) = a j + b j (x x j ) + c j (x x j ) 2 + d j (x x j ) 3 for x j x x j+1.): INPUT n; x 0, x 1,..., x n ; a 0 = f (x 0 ), a 1 = f (x 1 ),..., a n = f (x n ) OUTPUT a j, b j, c j, d j for j = 0, 1,..., n 1 Step 1 For i = 0, 1,..., n 1 set h i = x i+1 x i Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 9 / 29
19 Natural Cubic Spline Algorithm To construct the cubic spline interpolant S for the function f, defined at the numbers x 0 < x 1 < < x n, satisfying S (x 0 ) = S (x n ) = 0 (Note: S(x) = S j (x) = a j + b j (x x j ) + c j (x x j ) 2 + d j (x x j ) 3 for x j x x j+1.): INPUT n; x 0, x 1,..., x n ; a 0 = f (x 0 ), a 1 = f (x 1 ),..., a n = f (x n ) OUTPUT a j, b j, c j, d j for j = 0, 1,..., n 1 Step 1 Step 2 For i = 0, 1,..., n 1 set h i = x i+1 x i For i = 1, 2,..., n 1 set α i = 3 (a i+1 a i ) 3 (a i a i 1 ) h i h i 1 Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 9 / 29
20 Natural Cubic Spline Algorithm To construct the cubic spline interpolant S for the function f, defined at the numbers x 0 < x 1 < < x n, satisfying S (x 0 ) = S (x n ) = 0 (Note: S(x) = S j (x) = a j + b j (x x j ) + c j (x x j ) 2 + d j (x x j ) 3 for x j x x j+1.): INPUT n; x 0, x 1,..., x n ; a 0 = f (x 0 ), a 1 = f (x 1 ),..., a n = f (x n ) OUTPUT a j, b j, c j, d j for j = 0, 1,..., n 1 Step 1 Step 2 For i = 0, 1,..., n 1 set h i = x i+1 x i For i = 1, 2,..., n 1 set α i = 3 (a i+1 a i ) 3 (a i a i 1 ) h i h i 1 (Note: In what follows, Steps 3, 4, 5 and part of Step 6 solve a tridiagonal linear system using a Crout Factorization algorithm.) Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 9 / 29
21 Natural Cubic Spline Algorithm (Cont d) Step 3 Set l 0 = 1 µ 0 = 0 z 0 = 0 Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 10 / 29
22 Natural Cubic Spline Algorithm (Cont d) Step 3 Set l 0 = 1 µ 0 = 0 z 0 = 0 Step 4 For i = 1, 2,..., n 1 set l i = 2(x i+1 x i 1 ) h i 1 µ i 1 µ i = h i /l i z i = (α i h i 1 z i 1 )/l i Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 10 / 29
23 Natural Cubic Spline Algorithm (Cont d) Step 3 Set l 0 = 1 µ 0 = 0 z 0 = 0 Step 4 For i = 1, 2,..., n 1 set l i = 2(x i+1 x i 1 ) h i 1 µ i 1 µ i = h i /l i z i = (α i h i 1 z i 1 )/l i Step 5 Set l n = 1 z n = 0 c n = 0 Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 10 / 29
24 Natural Cubic Spline Algorithm (Cont d) Step 3 Set l 0 = 1 µ 0 = 0 z 0 = 0 Step 4 For i = 1, 2,..., n 1 set l i = 2(x i+1 x i 1 ) h i 1 µ i 1 µ i = h i /l i z i = (α i h i 1 z i 1 )/l i Step 5 Set l n = 1 z n = 0 c n = 0 Step 6 For j = n 1, n 2,..., 0 set c j = z j µ j c j+1 b j = (a j+1 a j )/h j h j (c j+1 + 2c j )/3 d j = (c j+1 c j )/(3h j ) Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 10 / 29
25 Natural Cubic Spline Algorithm (Cont d) Step 3 Set l 0 = 1 µ 0 = 0 z 0 = 0 Step 4 For i = 1, 2,..., n 1 set l i = 2(x i+1 x i 1 ) h i 1 µ i 1 µ i = h i /l i z i = (α i h i 1 z i 1 )/l i Step 5 Set l n = 1 z n = 0 c n = 0 Step 6 For j = n 1, n 2,..., 0 set c j = z j µ j c j+1 b j = (a j+1 a j )/h j h j (c j+1 + 2c j )/3 d j = (c j+1 c j )/(3h j ) Step 7 OUTPUT (a j, b j, c j, d j for j = 0, 1,..., n 1) & STOP Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 10 / 29
26 Outline 1 Unique natural cubic spline interpolant 2 Natural cubic spline approximating f (x) = e x 3 Natural cubic spline approximating 3 0 ex dx Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 11 / 29
27 Natural Spline Interpolant Example: f (x) = e x Use the data points (0, 1), (1, e), (2, e 2 ), and (3, e 3 ) to form a natural spline S(x) that approximates f (x) = e x. Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 12 / 29
28 Natural Spline Interpolant Example: f (x) = e x Use the data points (0, 1), (1, e), (2, e 2 ), and (3, e 3 ) to form a natural spline S(x) that approximates f (x) = e x. Solution (1/7) With n = 3, h 0 = h 1 = h 2 = 1 and the notation S(x) = S j (x) = a j + b j (x x j ) + c j (x x j ) 2 + d j (x x j ) 3 for x j x x j+1, Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 12 / 29
29 Natural Spline Interpolant Example: f (x) = e x Use the data points (0, 1), (1, e), (2, e 2 ), and (3, e 3 ) to form a natural spline S(x) that approximates f (x) = e x. Solution (1/7) With n = 3, h 0 = h 1 = h 2 = 1 and the notation S(x) = S j (x) = a j + b j (x x j ) + c j (x x j ) 2 + d j (x x j ) 3 for x j x x j+1, we have a 0 = 1, a 1 = e a 2 = e 2, a 3 = e 3 Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 12 / 29
30 Natural Spline Interpolant Solution (2/7) So the matrix A and the vectors b and x given in the Natural Spline Theorem See A, b & x have the forms c 0 A = b = 3(e 2 2e + 1) 3(e 3 2e 2 + e) and x = c 1 c c 3 Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 13 / 29
31 Natural Spline Interpolant Solution (2/7) So the matrix A and the vectors b and x given in the Natural Spline Theorem See A, b & x have the forms c 0 A = b = 3(e 2 2e + 1) 3(e 3 2e 2 + e) and x = c 1 c c 3 The vector-matrix equation Ax = b is equivalent to the system: c 0 = 0 c 0 + 4c 1 + c 2 = 3(e 2 2e + 1) c 1 + 4c 2 + c 3 = 3(e 3 2e 2 + e) c 3 = 0 Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 13 / 29
32 Natural Spline Interpolant Solution (3/7) This system has the solution c 0 = c 3 = 0 and, to 5 decimal places, c 1 = 1 5 ( e3 + 6e 2 9e + 4) c 2 = 1 5 (4e3 9e 2 + 6e 1) Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 14 / 29
33 Natural Spline Interpolant Solution (4/7) Solving for the remaining constants gives Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 15 / 29
34 Natural Spline Interpolant Solution (4/7) Solving for the remaining constants gives b 0 = 1 h 0 (a 1 a 0 ) h 0 3 (c 1 + 2c 0 ) = (e 1) 1 15 ( e3 + 6e 2 9e + 4) b 1 = 1 h 1 (a 2 a 1 ) h 1 3 (c 2 + 2c 1 ) = (e 2 e) 1 15 (2e3 + 3e 2 12e + 7) b 2 = 1 h 2 (a 3 a 2 ) h 2 3 (c 3 + 2c 2 ) = (e 3 e 2 ) 1 15 (8e3 18e e 2) Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 15 / 29
35 Natural Spline Interpolant Solution (5/7) and d 0 = 1 3h 0 (c 1 c 0 ) = 1 15 ( e3 + 6e 2 9e + 4) d 1 = 1 3h 1 (c 2 c 1 ) = 1 3 (e3 3e 2 + 3e 1) d 2 = 1 3h 2 (c 3 c 1 ) = 1 15 ( 4e3 + 9e 2 6e + 1) Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 16 / 29
36 Natural Spline Interpolant Solution (6/7) The natural cubic spine is described piecewise by x x 3 for x [0,1] S(x) = (x 1) (x 1) (x 1) 3 for x [1,2] (x 2) (x 2) (x 2) 3 for x [2,3] The spline and its agreement with f (x) = e x are as shown in the following diagram. Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 17 / 29
37 Natural Spline Interpolant Solution (7/7): Natural spline and its agreement with f (x) = e x e 3 y y = S(x) y = e x e 2 e x Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 18 / 29
38 Outline 1 Unique natural cubic spline interpolant 2 Natural cubic spline approximating f (x) = e x 3 Natural cubic spline approximating 3 0 ex dx Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 19 / 29
39 Natural Spline Interpolant Example: The Integral of a Spline Approximate the integral of f (x) = e x on [0, 3], which has the value 3 0 e x dx = e = , by piecewise integrating the spline that approximates f on this integral. Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 20 / 29
40 Natural Spline Interpolant Example: The Integral of a Spline Approximate the integral of f (x) = e x on [0, 3], which has the value 3 0 e x dx = e = , by piecewise integrating the spline that approximates f on this integral. Note: From the previous example, the natural cubic spine S(x) that approximates f (x) = e x on [0, 3] is described piecewise by x x 3 for x [0,1] S(x) = (x 1) (x 1) (x 1) 3 for x [1,2] (x 2) (x 2) (x 2) 3 for x [2,3] Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 20 / 29
41 Natural Spline Interpolant Solution (1/4) We can therefore write 3 0 S(x) = [ x x 3] dx [ (x 1) (x 1) (x 1) 3] dx [ (x 2) (x 2) (x 2) 3] dx Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 21 / 29
42 Natural Spline Interpolant Solution (2/4) Integrating and collecting values from like powers gives Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 22 / 29
43 Natural Spline Interpolant Solution (2/4) Integrating and collecting values from like powers gives 3 0 S(x) = [x x x 4 + [ (x 1) ] 1 0 (x 1)2 2 (x 1) (x 2)2 + [ (x 2) (x 2) ] 2 (x 1)4 4 1 ] 3 (x 2)4 4 2 Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 22 / 29
44 Natural Spline Interpolant Solution (3/4) Therefore: 3 0 S(x) = ( ) + 1 ( ) ( ) ( ) 4 = Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 23 / 29
45 Natural Spline Interpolant Solution (4/4) Because the nodes are equally spaced in this example the integral approximation is simply 3 0 S(x) dx = (a 0 + a 1 + a 2 ) (b 0 + b 1 + b 2 ) (c 0 + c 1 + c 2 ) (d 0 + d 1 + d 2 ) Numerical Analysis (Chapter 3) Cubic Spline Interpolation II R L Burden & J D Faires 24 / 29
46 Questions?
47 Reference Material
48 Cubic Spline Interpolant Definition Given a function f defined on [a, b] and a set of nodes a = x 0 < x 1 < < x n = b, a cubic spline interpolant S for f is a function that satisfies the following conditions: (a) S(x) is a cubic polynomial, denoted S j (x), on the subinterval [x j, x j+1 ] for each j = 0, 1,..., n 1; (b) S j (x j ) = f (x j ) and S j (x j+1 ) = f (x j+1 ) for each j = 0, 1,..., n 1; (c) S j+1 (x j+1 ) = S j (x j+1 ) for each j = 0, 1,..., n 2; (Implied by (b).) (d) S j+1 (x j+1) = S j (x j+1) for each j = 0, 1,..., n 2; (e) S j+1 j+1) = S j (x j+1 ) for each j = 0, 1,..., n 2; (f) One of the following sets of boundary conditions is satisfied: (i) S (x 0 ) = S (x n ) = 0 (natural (or free) boundary); (ii) S (x 0 ) = f (x 0 ) and S (x n ) = f (x n ) (clamped boundary).
49 Strictly Diagonally Dominant Matrices Theorem A strictly diagonally dominant matrix A is nonsingular. Moreover, in this case, Gaussian elimination can be performed on any linear system of the form Ax = b to obtain its unique solution without row or column interchanges, and the computations will be stable with respect to the growth of round-off errors. Return to Natural Spline Uniqueness Proof
50 Natural Spline Interpolant: Linear System Ax = b h 0 2(h 0 + h 1 ) h A = 0 h 1 2(h 1 + h 2 ) h hn 2 2(h n 2 + h n 1 ) h n h 1 (a 2 a 1 ) 3 h 0 (a 1 a 0 ) b =. 3 h n 1 (a n a n 1 ) 3 h n 2 (a n 1 a n 2 ) 0 c 0 c 1 and x =. c n Return to Natural Spline Example (f (x) = e x )
Parameterization. Michael S. Floater. November 10, 2011
Parameterization Michael S. Floater November 10, 2011 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to generate from point
February 2017 (1/20) 2 Piecewise Polynomial Interpolation 2.2 (Natural) Cubic Splines. MA378/531 Numerical Analysis II ( NA2 )
f f f f f (/2).9.8.7.6.5.4.3.2. S Knots.7.6.5.4.3.2. 5 5.2.8.6.4.2 S Knots.2 5 5.9.8.7.6.5.4.3.2..9.8.7.6.5.4.3.2. S Knots 5 5 S Knots 5 5 5 5.35.3.25.2.5..5 5 5.6.5.4.3.2. 5 5 4 x 3 3.5 3 2.5 2.5.5 5
APPM/MATH Problem Set 4 Solutions
APPM/MATH 465 Problem Set 4 Solutions This assignment is due by 4pm on Wednesday, October 16th. You may either turn it in to me in class on Monday or in the box outside my office door (ECOT 35). Minimal
Math 226A Homework 4 Due Monday, December 11th
Math 226A Homework 4 Due Monday, December 11th 1. (a) Show that the polynomial 2 n (T n+1 (x) T n 1 (x)), is the unique monic polynomial of degree n + 1 with roots at the Chebyshev points x k = cos ( )
Parameterization of triangular meshes
Parameterization of triangular meshes Michael S. Floater November 10, 2009 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to
Piecewise Polynomial Interpolation, cont d
Jim Lambers MAT 460/560 Fall Semester 2009-0 Lecture 2 Notes Tese notes correspond to Section 4 in te text Piecewise Polynomial Interpolation, cont d Constructing Cubic Splines, cont d Having determined
Mar. 20 Math 2335 sec 001 Spring 2014
Mar. 20 Math 2335 sec 001 Spring 2014 Chebyshev Polynomials Definition: For an integer n 0 define the function ( ) T n (x) = cos n cos 1 (x), 1 x 1. It can be shown that T n is a polynomial of degree n.
Homework #6 Brief Solutions 2012
Homework #6 Brief Solutions %page 95 problem 4 data=[-,;-,;,;4,] data = - - 4 xk=data(:,);yk=data(:,);s=csfit(xk,yk,-,) %Using the program to find the coefficients S =.456 -.456 -.. -.5.9 -.5484. -.58.87.
Cubic spline interpolation
Cubic spline interpolation In the following, we want to derive the collocation matrix for cubic spline interpolation. Let us assume that we have equidistant knots. To fulfill the Schoenberg-Whitney condition
Interpolation by Spline Functions
Interpolation by Spline Functions Com S 477/577 Sep 0 007 High-degree polynomials tend to have large oscillations which are not the characteristics of the original data. To yield smooth interpolating curves
Numerical Analysis I - Final Exam Matrikelnummer:
Dr. Behrens Center for Mathematical Sciences Technische Universität München Winter Term 2005/2006 Name: Numerical Analysis I - Final Exam Matrikelnummer: I agree to the publication of the results of this
Generalised Mean Averaging Interpolation by Discrete Cubic Splines
Publ. RIMS, Kyoto Univ. 30 (1994), 89-95 Generalised Mean Averaging Interpolation by Discrete Cubic Splines By Manjulata SHRIVASTAVA* Abstract The aim of this work is to introduce for a discrete function,
Linear Interpolating Splines
Jim Lambers MAT 772 Fall Semester 2010-11 Lecture 17 Notes Tese notes correspond to Sections 112, 11, and 114 in te text Linear Interpolating Splines We ave seen tat ig-degree polynomial interpolation
Chapter Introduction
Chapter 4.1 Introduction After reading this chapter, you should be able to 1. define what a matrix is. 2. identify special types of matrices, and 3. identify when two matrices are equal. What does a matrix
Polynomials tend to oscillate (wiggle) a lot, even when our true function does not.
AMSC/CMSC 460 Computational Methods, Fall 2007 UNIT 2: Spline Approximations Dianne P O Leary c 2001, 2002, 2007 Piecewise polynomial interpolation Piecewise polynomial interpolation Read: Chapter 3 Skip:
Natural Quartic Spline
Natural Quartic Spline Rafael E Banchs INTRODUCTION This report describes the natural quartic spline algorithm developed for the enhanced solution of the Time Harmonic Field Electric Logging problem As
LECTURE NOTES - SPLINE INTERPOLATION. 1. Introduction. Problems can arise when a single high-degree polynomial is fit to a large number
LECTURE NOTES - SPLINE INTERPOLATION DR MAZHAR IQBAL 1 Introduction Problems can arise when a single high-degree polynomial is fit to a large number of points High-degree polynomials would obviously pass
Convergence of C 2 Deficient Quartic Spline Interpolation
Advances in Computational Sciences and Technology ISSN 0973-6107 Volume 10, Number 4 (2017) pp. 519-527 Research India Publications http://www.ripublication.com Convergence of C 2 Deficient Quartic Spline
Numerical Methods in Physics Lecture 2 Interpolation
Numerical Methods in Physics Pat Scott Department of Physics, Imperial College November 8, 2016 Slides available from http://astro.ic.ac.uk/pscott/ course-webpage-numerical-methods-201617 Outline The problem
Math 225 Scientific Computing II Outline of Lectures
Math 225 Scientific Computing II Outline of Lectures Spring Semester 2003 I. Interpolating polynomials Lagrange formulation of interpolating polynomial Uniqueness of interpolating polynomial of degree
Discrete Cubic Interpolatory Splines
Publ RIMS, Kyoto Univ. 28 (1992), 825-832 Discrete Cubic Interpolatory Splines By Manjulata SHRIVASTAVA* Abstract In the present paper, existence, uniqueness and convergence properties of a discrete cubic
Homework #6 Brief Solutions 2011
Homework #6 Brief Solutions %page 95 problem 4 data=[-,;-,;,;4,] data = - - 4 xk=data(:,);yk=data(:,);s=csfit(xk,yk,-,) %Using the program to find the coefficients S =.456 -.456 -.. -.5.9 -.5484. -.58.87.
Spline Curves. Spline Curves. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1
Spline Curves Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1 Problem: In the previous chapter, we have seen that interpolating polynomials, especially those of high degree, tend to produce strong
Computational Physics PHYS 420
Computational Physics PHYS 420 Dr Richard H. Cyburt Assistant Professor of Physics My office: 402c in the Science Building My phone: (304) 384-6006 My email: rcyburt@concord.edu My webpage: www.concord.edu/rcyburt
Deficient Quartic Spline Interpolation
International Journal of Computational Science and Mathematics. ISSN 0974-3189 Volume 3, Number 2 (2011), pp. 227-236 International Research Publication House http://www.irphouse.com Deficient Quartic
Numerical Methods 5633
Numerical Methods 5633 Lecture 7 Marina Krstic Marinkovic mmarina@maths.tcd.ie School of Mathematics Trinity College Dublin Marina Krstic Marinkovic 1 / 10 5633-Numerical Methods Organisational To appear
More NP-complete Problems. CS255 Chris Pollett May 3, 2006.
More NP-complete Problems CS255 Chris Pollett May 3, 2006. Outline More NP-Complete Problems Hamiltonian Cycle Recall a hamiltonian cycle is a permutation of the vertices v i_1,, v i_n of a graph G so
INTERPOLATION BY QUARTIC SPLINE
INTERPOLATION BY QUARTIC SPLINE K.K.Nigam 1, Y.P. Dubey 2, Brajendra Tiwari 3, Anil Shukla 4 1 Mathematics Deptt., LNCT Jabalpur, MP, (India) 2 Sarswati Institution of Engg. & Technology, Jabalpur MP,
Multiple-Choice Test Spline Method Interpolation COMPLETE SOLUTION SET
Multiple-Choice Test Spline Method Interpolation COMPLETE SOLUTION SET 1. The ollowing n data points, ( x ), ( x ),.. ( x, ) 1, y 1, y n y n quadratic spline interpolation the x-data needs to be (A) equally
Lecture 8. Divided Differences,Least-Squares Approximations. Ceng375 Numerical Computations at December 9, 2010
Lecture 8, Ceng375 Numerical Computations at December 9, 2010 Computer Engineering Department Çankaya University 8.1 Contents 1 2 3 8.2 : These provide a more efficient way to construct an interpolating
Parametric Surfaces. Michael Kazhdan ( /657) HB , FvDFH 11.2
Parametric Surfaces Michael Kazhdan (601.457/657) HB 10.6 -- 10.9, 10.1 FvDFH 11.2 Announcement OpenGL review session: When: Wednesday (10/1) @ 7:00-9:00 PM Where: Olin 05 Cubic Splines Given a collection
Math Numerical Analysis
... Math 541 - Numerical Analysis Interpolation and Polynomial Approximation Piecewise Polynomial Approximation; Joseph M. Mahaffy, jmahaffy@mail.sdsu.edu Department of Mathematics and Statistics Dynamical
Exercise Set Decide whether each matrix below is an elementary matrix. (a) (b) (c) (d) Answer:
Understand the relationships between statements that are equivalent to the invertibility of a square matrix (Theorem 1.5.3). Use the inversion algorithm to find the inverse of an invertible matrix. Express
See the course website for important information about collaboration and late policies, as well as where and when to turn in assignments.
COS Homework # Due Tuesday, February rd See the course website for important information about collaboration and late policies, as well as where and when to turn in assignments. Data files The questions
CS 450 Numerical Analysis. Chapter 7: Interpolation
Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80
DM545 Linear and Integer Programming. Lecture 2. The Simplex Method. Marco Chiarandini
DM545 Linear and Integer Programming Lecture 2 The Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. 2. 3. 4. Standard Form Basic Feasible Solutions
10/26/ Solving Systems of Linear Equations Using Matrices. Objectives. Matrices
6.1 Solving Systems of Linear Equations Using Matrices Objectives Write the augmented matrix for a linear system. Perform matrix row operations. Use matrices and Gaussian elimination to solve systems.
lecture 10: B-Splines
9 lecture : -Splines -Splines: a basis for splines Throughout our discussion of standard polynomial interpolation, we viewed P n as a linear space of dimension n +, and then expressed the unique interpolating
Numerical Linear Algebra
Numerical Linear Algebra Probably the simplest kind of problem. Occurs in many contexts, often as part of larger problem. Symbolic manipulation packages can do linear algebra "analytically" (e.g. Mathematica,
3. Replace any row by the sum of that row and a constant multiple of any other row.
Math Section. Section.: Solving Systems of Linear Equations Using Matrices As you may recall from College Algebra or Section., you can solve a system of linear equations in two variables easily by applying
Interpolation and Splines
Interpolation and Splines Anna Gryboś October 23, 27 1 Problem setting Many of physical phenomenona are described by the functions that we don t know exactly. Often we can calculate or measure the values
Table for Third-Degree Spline Interpolation Using Equi-Spaced Knots. By W. D. Hoskins
MATHEMATICS OF COMPUTATION, VOLUME 25, NUMBER 116, OCTOBER, 1971 Table for Third-Degree Spline Interpolation Using Equi-Spaced Knots By W. D. Hoskins Abstract. A table is given for the calculation of the
On the Dimension of the Bivariate Spline Space S 1 3( )
On the Dimension of the Bivariate Spline Space S 1 3( ) Gašper Jaklič Institute of Mathematics, Physics and Mechanics University of Ljubljana Jadranska 19, 1000 Ljubljana, Slovenia Gasper.Jaklic@fmf.uni-lj.si
Lacunary Interpolation Using Quartic B-Spline
General Letters in Mathematic, Vol. 2, No. 3, June 2017, pp. 129-137 e-issn 2519-9277, p-issn 2519-9269 Available online at http:\\ www.refaad.com Lacunary Interpolation Using Quartic B-Spline 1 Karwan
The use of the Spectral Properties of the Basis Splines in Problems of Signal Processing
The use of the Spectral Properties of the Basis Splines in Problems of Signal Processing Zaynidinov Hakim Nasiritdinovich, MirzayevAvazEgamberdievich, KhalilovSirojiddinPanjievich Doctor of Science, professor,
An introduction to interpolation and splines
An introduction to interpolation and splines Kenneth H. Carpenter, EECE KSU November 22, 1999 revised November 20, 2001, April 24, 2002, April 14, 2004 1 Introduction Suppose one wishes to draw a curve
x n x n stepnumber k order r error constant C r+1 1/2 5/12 3/8 251/720 abs. stab. interval (α,0) /11-3/10
MATH 573 LECTURE NOTES 77 13.8. Predictor-corrector methods. We consider the Adams methods, obtained from the formula xn+1 xn+1 y(x n+1 y(x n ) = y (x)dx = f(x,y(x))dx x n x n by replacing f by an interpolating
99 International Journal of Engineering, Science and Mathematics
Journal Homepage: Applications of cubic splines in the numerical solution of polynomials Najmuddin Ahmad 1 and Khan Farah Deeba 2 Department of Mathematics Integral University Lucknow Abstract: In this
CS Elementary Graph Algorithms & Transform-and-Conquer
CS483-10 Elementary Graph Algorithms & Transform-and-Conquer Outline Instructor: Fei Li Room 443 ST II Office hours: Tue. & Thur. 1:30pm - 2:30pm or by appointments Depth-first Search cont Topological
A Modified Spline Interpolation Method for Function Reconstruction from Its Zero-Crossings
Scientific Papers, University of Latvia, 2010. Vol. 756 Computer Science and Information Technologies 207 220 P. A Modified Spline Interpolation Method for Function Reconstruction from Its Zero-Crossings
Problem #3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page Mark Sparks 2012
Problem # Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 490 Mark Sparks 01 Finding Anti-derivatives of Polynomial-Type Functions If you had to explain to someone how to find
Lecture 6: Chain rule, Mean Value Theorem, Tangent Plane
Lecture 6: Chain rule, Mean Value Theorem, Tangent Plane Rafikul Alam Department of Mathematics IIT Guwahati Chain rule Theorem-A: Let x : R R n be differentiable at t 0 and f : R n R be differentiable
Cubic Spline Questions
Cubic Spline Questions. Find natural cubic splines which interpolate the following dataset of, points:.0,.,.,.0, 7.0,.,.0,0.; estimate the value for. Solution: Step : Use the n- cubic spline equations
CS321 Introduction To Numerical Methods
CS3 Introduction To Numerical Methods Fuhua (Frank) Cheng Department of Computer Science University of Kentucky Lexington KY 456-46 - - Table of Contents Errors and Number Representations 3 Error Types
MATH 423 Linear Algebra II Lecture 17: Reduced row echelon form (continued). Determinant of a matrix.
MATH 423 Linear Algebra II Lecture 17: Reduced row echelon form (continued). Determinant of a matrix. Row echelon form A matrix is said to be in the row echelon form if the leading entries shift to the
For example, the system. 22 may be represented by the augmented matrix
Matrix Solutions to Linear Systems A matrix is a rectangular array of elements. o An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural of matrix) may be
Master Thesis. Comparison and Evaluation of Didactic Methods in Numerical Analysis for the Teaching of Cubic Spline Interpolation
Master Thesis Comparison and Evaluation of Didactic Methods in Numerical Analysis for the Teaching of Cubic Spline Interpolation Abtihal Jaber Chitheer supervised by Prof. Dr. Carmen Arévalo May 17, 2017
Solving Systems Using Row Operations 1 Name
The three usual methods of solving a system of equations are graphing, elimination, and substitution. While these methods are excellent, they can be difficult to use when dealing with three or more variables.
A Practical Review of Uniform B-Splines
A Practical Review of Uniform B-Splines Kristin Branson A B-spline is a convenient form for representing complicated, smooth curves. A uniform B-spline of order k is a piecewise order k Bezier curve, and
Friday, 11 January 13. Interpolation
Interpolation Interpolation Interpolation is not a branch of mathematic but a collection of techniques useful for solving computer graphics problems Basically an interpolant is a way of changing one number
X 1 = = 1 2, X 2 = = 1 4.
Solutions: 5: Choose an initial value for X and use values of r that are less than, greater than, and equal to one to test the above statements by computing two values of X for each value of r Choose X,
Lecture 2 September 3
EE 381V: Large Scale Optimization Fall 2012 Lecture 2 September 3 Lecturer: Caramanis & Sanghavi Scribe: Hongbo Si, Qiaoyang Ye 2.1 Overview of the last Lecture The focus of the last lecture was to give
Spline fitting tool for scientometric applications: estimation of citation peaks and publication time-lags
Spline fitting tool for scientometric applications: estimation of citation peaks and publication time-lags By: Hicham Sabir Presentation to: 21 STI Conference, September 1th, 21 Description of the problem
Methods of solving sparse linear systems. Soldatenko Oleg SPbSU, Department of Computational Physics
Methods of solving sparse linear systems. Soldatenko Oleg SPbSU, Department of Computational Physics Outline Introduction Sherman-Morrison formula Woodbury formula Indexed storage of sparse matrices Types
Consider functions such that then satisfies these properties: So is represented by the cubic polynomials on on and on.
1 of 9 3/1/2006 2:28 PM ne previo Next: Trigonometric Interpolation Up: Spline Interpolation Previous: Piecewise Linear Case Cubic Splines A piece-wise technique which is very popular. Recall the philosophy
Interpolation. Chapter Information from Data
Chapter 5 Interpolation 5. Information from Data The topic of this chapter is interpolation, which relates to passing a curve through a set of data points. To put this in context, extracting information
Piecewise polynomial interpolation
Chapter 2 Piecewise polynomial interpolation In ection.6., and in Lab, we learned that it is not a good idea to interpolate unctions by a highorder polynomials at equally spaced points. However, it transpires
Connected Minimal Acceleration Trigonometric Curves
Connected Minimal Acceleration Trigonometric Curves Tony Barrera Barrera Kristiansen AB Anders Hast Creative Media Lab, University of Gävle Ewert Bengtsson Centre for Image Analysis Uppsala University
Lecture 2.2 Cubic Splines
Lecture. Cubic Splines Cubic Spline The equation for a single parametric cubic spline segment is given by 4 i t Bit t t t i (..) where t and t are the parameter values at the beginning and end of the segment.
60 2 Convex sets. {x a T x b} {x ã T x b}
60 2 Convex sets Exercises Definition of convexity 21 Let C R n be a convex set, with x 1,, x k C, and let θ 1,, θ k R satisfy θ i 0, θ 1 + + θ k = 1 Show that θ 1x 1 + + θ k x k C (The definition of convexity
Matrix Multiplication and All Pairs Shortest Paths (2002; Zwick)
Matrix Multiplication and All Pairs Shortest Paths (2002; Zwick) Tadao Takaoka, University of Canterbury www.cosc.canterbury.ac.nz/tad.takaoka INDEX TERMS: all pairs shortest path problem, matrix multiplication,
Outline. CS38 Introduction to Algorithms. Linear programming 5/21/2014. Linear programming. Lecture 15 May 20, 2014
5/2/24 Outline CS38 Introduction to Algorithms Lecture 5 May 2, 24 Linear programming simplex algorithm LP duality ellipsoid algorithm * slides from Kevin Wayne May 2, 24 CS38 Lecture 5 May 2, 24 CS38
4 Integer Linear Programming (ILP)
TDA6/DIT37 DISCRETE OPTIMIZATION 17 PERIOD 3 WEEK III 4 Integer Linear Programg (ILP) 14 An integer linear program, ILP for short, has the same form as a linear program (LP). The only difference is that
Quadratic and cubic b-splines by generalizing higher-order voronoi diagrams
Quadratic and cubic b-splines by generalizing higher-order voronoi diagrams Yuanxin Liu and Jack Snoeyink Joshua Levine April 18, 2007 Computer Science and Engineering, The Ohio State University 1 / 24
Interpolation - 2D mapping Tutorial 1: triangulation
Tutorial 1: triangulation Measurements (Zk) at irregular points (xk, yk) Ex: CTD stations, mooring, etc... The known Data How to compute some values on the regular spaced grid points (+)? The unknown data
A Random Variable Shape Parameter Strategy for Radial Basis Function Approximation Methods
A Random Variable Shape Parameter Strategy for Radial Basis Function Approximation Methods Scott A. Sarra, Derek Sturgill Marshall University, Department of Mathematics, One John Marshall Drive, Huntington
Computational Methods CMSC/AMSC/MAPL 460. Vectors, Matrices, Linear Systems, LU Decomposition, Ramani Duraiswami, Dept. of Computer Science
Computational Methods CMSC/AMSC/MAPL 460 Vectors, Matrices, Linear Systems, LU Decomposition, Ramani Duraiswami, Dept. of Computer Science Zero elements of first column below 1 st row multiplying 1 st
Computation of interpolatory splines via triadic subdivision
Computation of interpolatory splines via triadic subdivision Valery A. Zheludev and Amir Z. Averbuch Abstract We present an algorithm for computation of interpolatory splines of arbitrary order at triadic
2.4. A LIBRARY OF PARENT FUNCTIONS
2.4. A LIBRARY OF PARENT FUNCTIONS 1 What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal function. Identify and graph step and
Curve fitting using linear models
Curve fitting using linear models Rasmus Waagepetersen Department of Mathematics Aalborg University Denmark September 28, 2012 1 / 12 Outline for today linear models and basis functions polynomial regression
Mutation-linear algebra and universal geometric cluster algebras
Mutation-linear algebra and universal geometric cluster algebras Nathan Reading NC State University Mutation-linear ( µ-linear ) algebra Universal geometric cluster algebras The mutation fan Universal
UNIVERSITY OF CALGARY. Subdivision Surfaces. Advanced Geometric Modeling Faramarz Samavati
Subdivision Surfaces Surfaces Having arbitrary Topologies Tensor Product Surfaces Non Tensor Surfaces We can t find u-curves and v-curves in general surfaces General Subdivision Coarse mesh Subdivision
Element Quality Metrics for Higher-Order Bernstein Bézier Elements
Element Quality Metrics for Higher-Order Bernstein Bézier Elements Luke Engvall and John A. Evans Abstract In this note, we review the interpolation theory for curvilinear finite elements originally derived
CHAPTER 8. Copyright Cengage Learning. All rights reserved.
CHAPTER 8 RELATIONS Copyright Cengage Learning. All rights reserved. SECTION 8.3 Equivalence Relations Copyright Cengage Learning. All rights reserved. The Relation Induced by a Partition 3 The Relation
(Spline, Bezier, B-Spline)
(Spline, Bezier, B-Spline) Spline Drafting terminology Spline is a flexible strip that is easily flexed to pass through a series of design points (control points) to produce a smooth curve. Spline curve
Engineering Analysis ENG 3420 Fall Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00
Engineering Analysis ENG 3420 Fall 2009 Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00 1 Lecture 24 Attention: The last homework HW5 and the last project are due on Tuesday November
Spline Models. Introduction to CS and NCS. Regression splines. Smoothing splines
Spline Models Introduction to CS and NCS Regression splines Smoothing splines 3 Cubic Splines a knots: a< 1 < 2 < < m
On the maximum rank of completions of entry pattern matrices
Linear Algebra and its Applications 525 (2017) 1 19 Contents lists available at ScienceDirect Linear Algebra and its Applications wwwelseviercom/locate/laa On the maximum rank of completions of entry pattern
4.2 and 4.6 filled in notes.notebook. December 08, Integration. Copyright Cengage Learning. All rights reserved.
4 Integration Copyright Cengage Learning. All rights reserved. 1 4.2 Area Copyright Cengage Learning. All rights reserved. 2 Objectives Use sigma notation to write and evaluate a sum. Understand the concept
Matrices and Systems of Linear Equations
Chapter The variable x has now been eliminated from the first and third equations. Next, we eliminate x3 from the first and second equations and leave x3, with coefficient, in the third equation: System:
Iterative Methods for Linear Systems
Iterative Methods for Linear Systems 1 the method of Jacobi derivation of the formulas cost and convergence of the algorithm a Julia function 2 Gauss-Seidel Relaxation an iterative method for solving linear
Math 2331 Linear Algebra
4.2 Null Spaces, Column Spaces, & Linear Transformations Math 233 Linear Algebra 4.2 Null Spaces, Column Spaces, & Linear Transformations Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu
Surface Approximation and Interpolation via Matrix SVD
Surface Approximation and Interpolation via Matrix SVD Andrew E. Long and Clifford A. Long Andy Long (longa@nku.edu) is at Northern Kentucky University. He received his Ph.D. in applied mathematics at
Moving Beyond Linearity
Moving Beyond Linearity Basic non-linear models one input feature: polynomial regression step functions splines smoothing splines local regression. more features: generalized additive models. Polynomial
Iterative Algorithms I: Elementary Iterative Methods and the Conjugate Gradient Algorithms
Iterative Algorithms I: Elementary Iterative Methods and the Conjugate Gradient Algorithms By:- Nitin Kamra Indian Institute of Technology, Delhi Advisor:- Prof. Ulrich Reude 1. Introduction to Linear
Chapter 1 BACKGROUND
Chapter BACKGROUND. Introduction In many areas of mathematics and in applications of mathematics, it is often necessary to be able to infer information about some function using only a limited set of sample
Interactive Graphics. Lecture 9: Introduction to Spline Curves. Interactive Graphics Lecture 9: Slide 1
Interactive Graphics Lecture 9: Introduction to Spline Curves Interactive Graphics Lecture 9: Slide 1 Interactive Graphics Lecture 13: Slide 2 Splines The word spline comes from the ship building trade
Contents. I The Basic Framework for Stationary Problems 1
page v Preface xiii I The Basic Framework for Stationary Problems 1 1 Some model PDEs 3 1.1 Laplace s equation; elliptic BVPs... 3 1.1.1 Physical experiments modeled by Laplace s equation... 5 1.2 Other
Almost Curvature Continuous Fitting of B-Spline Surfaces
Journal for Geometry and Graphics Volume 2 (1998), No. 1, 33 43 Almost Curvature Continuous Fitting of B-Spline Surfaces Márta Szilvási-Nagy Department of Geometry, Mathematical Institute, Technical University
SPLINE APPROXIMATION VIA THE CONTROL POLYGON
SPLINE APPROXIMATION VIA THE CONTROL POLYGON by Håkon Mørk THESIS for the degree of MASTER S DEGREE IN APPLIED MATHEMATICS AND MECHANICS (Master i Anvendt matematikk og mekanikk) Faculty of Mathematics